Such a closure device is known from EP 0 956 088 B1. The known closure device is configured as a medical valve for controlling the flow of fluid between a medical instrument and a catheter tip of a catheter. The valve has a support housing with a connection piece, the fluid passage of which can be closed by means of an elastically deformable closure membrane. On the basis of FIGS. 28 and 29, a dual valve is also presented in EP 0 956 088 B1, two elastically deformable closure membranes being arranged in series one behind the other in the support housing. The first closure membrane, closing the fluid passage, is opened by the insertion of a syringe or some other Luer connection part. The second closure membrane, at a distance in the direction of the fluid, is opened by corresponding liquid pressure, which occurs by introducing a fluid from the syringe or the Luer connection part into the support housing of the closure device.